from platonic solids to quivers
TRANSCRIPT
From Platonic solids to quivers
Gwyn Bellamy
www.maths.gla.ac.uk/∼gbellamy
June 21, 2016
Abstract
This course will be a whirlwind tour through representation theory, a major branch
of modern algebra. We being by considering the symmetry groups of the Platonic solids,
which leads naturally to the notion of a reflection group and its associated root system.
The classification of these reflection groups gives us our first examples of quivers (= direct
graphs). Though easy to define, we’ll see that the representation theory associated to quivers
is very rich. We will use quivers to illustrate the key concepts, ideas and problems that
appear throughout representation theory. Coming full circle, the course will culminate with
the beautiful theorem by Gabriel, classifying the quivers of finite type in terms of the root
systems of reflection groups. The ultimate goal of the course is to give students a glimpse of
the beauty and unity of this field of research, which is today very active in the U.K.
Contents
1 The Platonic solids 1
2 Reflection groups and root systems 10
3 Quivers 27
4 Gabriel’s Theorem 34
5 Appendix 48
Introduction
This series of four lectures is designed to be a crash course in representation theory; the only back-
ground knowledge assumed is basic group theory and linear algebra. We start with the ancient
Greeks and their study of the Platonic solids and end up finally with Gabriel’s theorem classifying
quivers with finitely many indecomposable representations. At first glance, the topics covered -
the Platonic solids, finite reflection groups and quiver, seem completely unrelated. But one of
the key messages of the lectures is that there is in fact a deep connection between these objects.
Underlying each of their classifications is a certain graph called a Coxeter/Euler graph, and the
corresponding positive definite symmetric form. Remarkably, these graphs appear in several other
areas of mathematics; in section 4.6 I have listed all those that I know, but I’m certain that there
are several more.
As the name suggests, the Platonic solids were well-known to the ancient Greeks. There seems
to some contention as to who first discovered them (some attribute this to Pythagoras, others to
Theaetetus) but they were popularized by Plato, who associated to each of the first four solids
one of the four elements, “fundamental building blocks” of the universe,
tetrahedron↔ fire
cube↔ earth
octahedron↔ air
icosahedron↔ water
The dodecahedron was meant to represent the whole universe. Later the German astronomer
Johannes Kepler based his model of the solar system on the Platonic solids. For a given solid P
there is a unique sphere P ⊂ S such that all the vertices of P lie on S. There is a second, unique,
sphere S ′ ⊂ P such that the midpoint of each face of P lies in S ′. By nesting the Platonic solids
P1 ⊂ P2 ⊂ · · · ⊂ P5 in the correct way, we get six concentric spheres S1 ⊂ · · · ⊂ S6. Each of these
six spheres was meant to correspond to the orbit of the six planets that were known at that time,
so that each solid separates a pair of planets,
octahedron↔ Mercury and Venus
icosahedron↔ Venus and Earth
dodecahedron↔ Earth and Mars
tetrahedron↔ Mars and Jupiter
cube↔ Jupiter and Saturn;
see Figure 1.1. Though we know today that this model can’t possibly have any real meaning, it is
remarkable how close the fit actually was. In the first lecture we will focus on the symmetries of
the Platonic solids. Due to the mysterious duality that the solids posses, these symmetry groups
are relatively easy to calculate. Remarkably, these groups are all reflection groups i.e. they are
generated by reflections.
In the second lecture, we study in more detail this notion of a reflection group. It is a classical
result that one can classify all finite reflection groups. This classification result will be at the heart
of the lecture. In order to state it and outline a proof, we will be forced to introduce Coxeter
graphs and their corresponding symmetric bilinear forms. The fact that one can classify finite
reflection groups relies on the fact that one can classify those Coxeter graphs that are positive
definite. To go between reflection groups and Coxeter graphs, one needs the notion of a root
system. This is a very natural object that one can associate to a reflection group, but it still
contains all the information needed to reconstruct the group. For a given root system, it is easy
to work out what the corresponding Coxeter graph is.
In the third lecture we leave behind Platonic solids and reflection groups and turn instead to
quivers. A quiver is simply a directed graph. These were first introduced by Gabriel to encode
in a very simple, combinatorial way, very difficult problems in linear algebra. This gives rise to
the notion of a representation of a quiver. Though the definition is very simple, representations
can turn out to have very complex structures. As such they are still the focus of much research
by mathematicians today. Quivers are also the ideal introduction to representation theory - this
major branch of algebra aims to understand objects, whether rings, groups, Lie algebras,.... by
understanding how they act on vector spaces. Due to their combinatorial natural, Quiver represen-
tations provide a large source of interesting examples of representations and are useful in testing
ii
conjectures in the theory. They also provide us with the ideal setting in which to introduce the
basic concepts of the subject such as a simple or indecomposable representation, and the notion
of semi-simplicity.
In the final lecture we will focus on one of the fundamental results in the theory of quiver
representations, that of Gabriel’s Theorem. It’s easy to see that in order to classify all the rep-
resentations of a quiver it suffices to classify those that are indecomposable i.e. those that we
cannot decompose into smaller representations. Therefore it is natural to ask for which quiv-
ers might we expect there to be only finitely many indecomposable representations. Gabriel’s
Theorem provides a complete answer to this question - it is precisely those quivers whose un-
derlying graph is a positive definite Euler graph. This provides us (amazingly!) with a link to
the second lecture on reflections groups, since the positive definite Euler graphs are exactly the
positive definite Coxeter graphs with at most one edge between any two vertices. Not only that,
Gabriel showed that there is a bijection between the indecomposables of the quiver and the roots
in the corresponding root system. Thus, a concept introduced in the theory of reflection groups
appears mysteriously in answering one of the basic questions in quiver representations. In fact,
it turns out that root systems play an important role throughout the representations theory of
quivers. This is exemplified by Kac’s Theorem, which is a vast generalization of Gabriel’s Theorem.
At the end of each section I have listed several sources where you can learn more about the
topic under consideration. I hope the lecture notes will persuade you to dip into some of these.
The reward for those of you who do so is the revelation of a beautiful, rich world which touches on
many of the cornerstones of mathematics, such as algebraic geometry, topology, group theory and,
of course, representation theory. Here I would just like to mention the survey [19] by Slodowy and
the thesis [21] by van Hoboken, which are both excellent expositions of the connection between
Platonic solids, finite subgroups of SU(2) and Lie algebras of simply laced type.
1
Figure 1: The five Platonic solids.
1 The Platonic solids
In the first lecture, we travel all the way back to ancient Greece, and explore the Platonic solids
that so fascinated the Greeks at that time (so much so that Plato saw them as the very building
blocks of the universe).
1.1 The five regular solids
A convex regular polyhedron in R3 is called a Platonic solid or regular solid. Here, a polyhedron
is a region bounded by planes in R3. It has two-dimensional faces which meet in one-dimensional
edges, which meet in vertices. We shall assume that a polyhedron is bounded. This implies that
it is compact as a topological space. We say that a polyhedron is regular if all its faces, edges
and vertices are equal. That is, all the faces meet at the same angle and that the same number of
edges meet at the same angles at each vertex. In particular, this implies that all the faces are the
same regular polygon.
As the name suggests, the Platonic solids were popularized by Plato, therefore they are also
know as the Platonic solids. In Figure 1.1, you’ll see some Platonic solids. The Schlafli symbol
symbol of a Platonic solid is {p, q}, where p is the number of faces meeting at a given vertex, each
of which is a q-gon. A moment’s thought should convince you that there is at most one Platonic
solid with Schlafli symbol {p, q}.
Theorem 1.1. The five polygons of Figure 1.1 are the only regular polygons.
Proof. Let {p, q} be the Schlafli symbol of the polygon. Then we have p faces (each a regular
q-gon) meeting at every vertex. The angle at the corner of the q-gon is (q − 2)π/q and since the
polyhedron is convex we must have p×(q−2)π/q < 2π, which implies that (p−2)(q−2) < 4. The
1
only positive integers satisfying this are (q, p) = (3, 3), (4, 3), (3, 4), (5, 3), (3, 5), corresponding to
the above five.
Here’s some more information about them:
Polyhedron Vertices V Edges E Faces F Schlafli symbol
Tetrahedron T 4 6 4 {3, 3}
Hexahedron H 8 12 6 {4, 3}
Octahedron O 6 12 8 {3, 4}
Dodecahedron D 20 30 12 {5, 3}
Icosahedron I 12 30 20 {3, 5}
If you’re curious to know, “hedron” is derived from the Greek for face. From now on, we’ll call
the Hexahedron the “cube” for simplicity.
1.2 Topology
Notice that the numbers (V,E, F ) satisfy a remarkable property, namely we always have
V − E + F = 2.
Euler showed that this equation actually holds for any convex polyhedron in R3. This is because
V − E + F is a topological property of the shapes. Topologically, any convex polyhedron is
homeomorphic to the 3-sphere. Thus, the quantity V − E + F will always equal 2, which is the
Euler characteristic of the 3-sphere. More generally, for any bounded, closed polyhedron P in R3,
2
Figure 2: Kepler’s model of the solar system, based on the five Platonic solids.
3
we have
V − E + F = 2− 2g, (1)
where g is the genus of P ; this is the number of “holes” in P .
Exercise 1.2. Check that equation (1) holds for the Platonic solids and for the torus with trian-
gulation
where one should glue together the two edges labeled by a single arrow and similarly for the two
arrows labeled by a double arrow.
1.3 Duality
The Platonic solids possess a beautiful duality - take a solid P and construct a new polyhedron
Q within P as follows. For each face of P put a vertex at the centre of this face. These will be
the vertices of Q. If two faces of P share a common edge then draw a new edge from the vertex
at the centre of one face to the vertex at the centre of the other face. These will be the edges of
Q. Notice that there are bijections
{ faces of P } 1:1←→ { vertices of Q },
{ edges of P } 1:1←→ { edges of Q }.
Finally, for each vertex of P , form a face of Q by filling in the polygon bounded by all the edges
of Q that correspond to the edges of P meeting at that vertex. This gives the final bijection,
{ vertices of P } 1:1←→ { faces of Q }.
We say that Q is the dual of P . Repeating, the dual of Q is just a smaller copy of P sitting in Q
(sitting in P ...).
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Exercise 1.3. If the Schlafli symbol of P is {p, q}, check that the Schlafli symbol of its dual Q is
{q, p}.
Figure 1.3 shows the dual Platonic solids. We see that the tetrahedron is self-dual, the cube
and the octahedron are dual to one another, as are the dodecahedron and icosahedron.
1.4 Symmetries
In order to describe the symmetry groups of the Platonic solids, we begin with some general
observations. The group of symmetries of P will be denoted W (P ), a subgroup of GL(Rn).
1. If V is the set of vertices of P then the symmetries of P permute the elements in V . This
defines a homomorphism W (P ) → Sym(V ). It is easy to see that this is injective. Since
Sym(V ) is a finite group, this implies that W (P ) is a finite group.
2. W (P ) is a finite subgroup of O(3,R); see Proposition 5.3 of the appendix.
3. This implies that det(g) = ±1 for each g ∈ W (P ). In the case det(g) = 1, g ∈ SO(3,R) is
a rotation of R3. We include a proof of this fact in section 5.2 of the appendix.
4. If the map −IdR3 belongs to W (P ) then notice that det(g) = −1 if and only if det(−g) = 1.
This implies that W (P ) = W0(P )×{±IdR3}, where W0(P ) = W (P )∩SO(3,R) is the group
of rotational symmetries of P .
5. Dual solids have the same symmetry group.
Since there are only three dual pairs, (5) means that there are only three symmetry groups to
compute.
The symmetries of the tetrahedron
The tetrahedron T has 4 vertices. Therefore we have an embedding W (T) ↪→ S4, where S4 is the
symmetric group on four letters. However, it is clear that for any pair of vertices {a, b} there is a
reflection of T that swaps a and b and fixes the other two vertices. Since S4 is generated by such
transpositions, this implies that the embedding is an isomorphism. Notice that this also implies
that W (T) is generated by reflections.
5
Figure 3: Dual pairs of Platonic solids.
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The symmetries of the cube/octahedron
We compute the symmetries of the cube. In this case −IdR3 acts on H. Therefore, it suffices to
describe the rotational symmetries of the cube. Any rotation of H will have an axis of rotation.
(A) The axis of rotation passes through the centre of a pair of opposite faces. For a given pair
of faces, there are three non-trivial rotations, giving 3× 3 = 9 non-trivial rotations.
(B) The axis of rotation passes through the centre of a pair of opposite edges. For a given pair
of edges, there is only one non-trivial rotation, giving 12/2 = 6 non-trivial rotations in total.
(C) The axis of rotation passes through the centre of a pair of opposite vertices. For a given pair
of vertices, there are two non-trivial rotations, giving 2× (8/2) = 8 non-trivial rotations.
Hence, if we remember to include the identity, we have
|W0(P )| = 1 + 9 + 6 + 8 = 24, |W (P )| = 48.
In fact, W0(P ) is also isomorphic to S4. To see this, let D = {`1, . . . , `4} be the set of lines
connecting opposite vertices in the cube. Then W0(P ) permutes the elements of D and we get a
morphism W0(P )→ S4, which one can check is an isomorphism.
The symmetries of the dodecahedron/icosahedron
We compute the symmetries of the dodecahedron. As before, −IdR3 acts on D. Therefore, it
suffices to describe the rotational symmetries of the dodecahedron, just as we did for the cube.
Any rotation of D will have an axis of rotation.
(A) The axis of rotation passes through the centre of a pair of opposite faces. For a given pair
of faces, there are four non-trivial rotations, giving 4× (12/2) = 24 non-trivial rotations.
(B) The axis of rotation passes through the centre of a pair of opposite edges. For a given pair of
edges, there is only one non-trivial rotation, giving 30/2 = 15 non-trivial rotations in total.
(C) The axis of rotation passes through the centre of a pair of opposite vertices. For a given pair
of vertices, there are two non-trivial rotations, giving 2× (20/2) = 20 non-trivial rotations.
Hence, if we remember to include the identity, we have
|W0(P )| = 1 + 24 + 15 + 20 = 60, |W (P )| = 120.
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In fact, as abstract groups, one can show that W0(P ) is isomorphic to the alternating group A5;
see [6, page 50].
1.5 Reflection groups
The symmetry groups of the Platonic solids all have one key property in common. They are all
examples of finite reflection groups. We denote by O(n,R) the group of orthogonal n×n matrices
{A | AAT = 1}. Equivalently, this is the group of all angle and length preserving transformations
of Rn i.e.
g ∈ O(3,R) ⇔ (g · x, g · y) = (x, y) ∀ x, y ∈ Rn.
A reflection on Rn is a linear map s ∈ O(n,R) such that
1. dim FixRn(s) = n− 1;
2. s2 = id.
Here FixRn(s) = {x ∈ Rn | s(x) = x} is a vector subspace of Rn.
Definition 1.4. A finite subgroup W of GL(Rn) is said to be a reflection group if it is generated
by a set S = {s1, . . . , sk} of Rn. That is, W is the smallest subgroup of GL(Rn) containing all the
reflections in S.
Notice that if W is a reflection group then W ⊂ O(n,R) i.e. each element in W preserves both
angles and lengths.
Exercise 1.5. Using the fact that g ∈ W (P ) is a reflection if and only if it has one eigenvalue equal
to −1 and two eigenvalues equal to 1, count the number of reflections in W (H) and W (D).
We finish the lecture with a key example of a reflection group. Generalizing the tetrahedron,
the n-simplex is defined to be
∆n ={
(t1, . . . , tn+1) ∈ Rn+1 | t1 + · · ·+ tn+1 = 1, ti ≥ 0 ∀ i},
so that ∆2 is the regular triangle and ∆3 is the tetrahedron. The vertices of ∆n are
e1 = (1, 0, . . . ), e2 = (0, 1, 0, . . . ), . . . ,
of which there are n + 1 in total. Therefore the action of the symmetry group W (∆n) of the n-
simplex on its set of vertices defines an embedding W (∆n) ↪→ Sn+1. On the other hand, consider
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the transformation si,j of Rn which swaps the ith and jth coordinate. Clearly s2i,j = 1 and the
subspace Fix(si,j) has basis {tk | k 6= i, j} ∪ {ti + tj} and hence is (n− 1)-dimensional. Thus, si,j
is a reflection. We see that si,j just swaps the ith and jth vertex of ∆n i.e. the image of si,j in
Sn+1 is the transposition (i, j). Hence, we have shown
Proposition 1.6. The symmetry group W (∆n) is a reflection group, isomorphic to the symmetric
group Sn+1.
1.6 Remarks
The classical reference on Platonic solids and their symmetry groups has to be the book Reg-
ular Polytopes, by H.S.M. Coxeter [6]. The lecture notes [19] by Slodowy are also a fantastic
introduction to these classical objects, but from the modern point of view. Finally, I should
mention the superb thesis [21] by van Hoboken, where I first learned about these things. See
http://www.math.ucr.edu/home/baez/FUN.html for a great deal more on the Platonic solids
and much else besides.
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2 Reflection groups and root systems
Recall that we ended the first lecture with the definition of a (finite) reflection group: a finite
subgroup W of GL(Rn) is said to be a reflection group if it is generated by a set S = {s1, . . . , sk}of reflections of Rn. That is, si : Rn → Rn is a orthogonal linear map fixing a hyperplane H ⊂ Rn
and s2i = 1. This means that
si(x) = x− 2(x, α)
(α, α)α, (2)
for some (in fact any) vector α perpendicular to H. We’ll call α the root of si and write si = sα.
Our presentation in this lecture is based on the book [12] by Humphreys.
2.1 Root systems
In order to classify the finite reflection groups, we introduce the notion of a root system. We fix
a real vector space Rn with a positive definite symmetric bilinear form (−,−).
Exercise 2.1. Using equation (2) show that (w · u, v) = (u,w−1 · v) for all u, v ∈ Rn and w ∈ W .
Hint: first check for w = s, a reflection, then use the fact that every element in W can be written
as a product of reflections.
For simplicity, we will assume throughout that (Rn)W = {v ∈ Rn | w · v = v, ∀ w ∈ W} equals
{0}. Otherwise, we simply replace Rn by the orthogonal compliment V = {v ∈ Rn | (v, u) =
0, ∀ u ∈ (Rn)W} to (Rn)W in Rn. The subspace V is W -stable.
Definition 2.2. A finite subset R of Rn is called a root system if
(R0) R spans Rn.
(R1) If α ∈ R then the only multiplies of α in R are ±α.
(R2) If α ∈ R then the reflection sα maps R to itself.
The reflection groupW (R) associated to R is the subgroup ofGL(Rn) generated by all {sα | α ∈R}. We give a couple of examples of root systems and the corresponding reflection groups.
Example 2.3. The dihedral groups. The dihedral group I2(m) of order 2m is the group of sym-
metries of the m-gon in R2. Taking every root to have length one, the corresponding root system
is
R =
{(cos
2kπ
m, sin
2kπ
m
)| k = 0, 1, . . . ,m− 1
}.
When m = 6, we get:
10
α
β
Here ∆ = {α, β} is a set of simple roots for R.
Example 2.4. The symmetric group. Let
E =
{x =
n+1∑i=1
xiεi ∈ Rn+1 |n+1∑i=1
xi = 0
}' Rn,
where {ε1, . . . , εn+1} is the standard basis of Rn+1 with (εi, εj) = δi,j. Let R = {εi − εj | 1 ≤ i 6=j ≤ n+ 1}. Then, it is an exercise to check that R is a root system for Sn.
Lemma 2.5. Let R be a root system. Then the reflection group W (R) of R is finite.
Proof. By axiom (R2) of a root system every sα maps the finite set R into itself. Therefore, there
is a group homomorphism W → SN , where N = |R|. This map is injective: if w ∈ W such that
its action on R is trivial then, since R spas Rn by (R0), w acts trivially on Rn i.e. w = 1.
Choose a vector v ∈ Rn such that (v, α) 6= 0 for all α ∈ R. Then either α ∈ R+ := {α ∈R | (v, α) > 0} or α ∈ R− := −R+ i.e. we get a partition R = R+ tR− into positive and negative
roots.
Definition 2.6. Let R = R+ ∪R− be a root system, partitioned into positive and negative roots.
A set of simple roots for R is a subset ∆ ⊂ R+ such that
(S1) ∆ is a basis of Rn.
(S2) Each β ∈ R+ can be written as β =∑
α∈∆ mαα, where mα ≥ 0 for all α.
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It is also possible to define a set of simple roots, without first choosing a partition of R into
positive and negative roots. Replace the axiom (S2) by
(S2’) Each β ∈ R can be written as β =∑
α∈∆ mαα, where all mα ≥ 0 or all mα ≤ 0.
Then a set of simple roots automatically defines a partition of R; a root β is positive if and only
if all mα ≥ 0. The problem with the above definition is that it is not at all clear that a given root
system contains a set of simple roots. However, one can show:
Theorem 2.7. Let R be a root system.
1. There exists a set ∆ of simple roots in R.
2. The group W is generated by S := {sα | α ∈ ∆}.
3. For any two sets of simple roots ∆,∆′, there exists w ∈ W such that w(∆) = ∆′.
4. For α, β ∈ ∆, α 6= β we have (α, β) ≤ 0.
In fact the reflection group W acts simply transitively on the collections of all sets of simple
roots. The construction of all ∆’s is quite easy, the difficulty is in showing that the sets constructed
are indeed sets of simple roots and that the properties of Theorem 2.7 are satisfied.
Given a set of simple reflections, we call the corresponding set of reflections S a set of simple
reflections for W . Now that we have a choice of generators for W , we can ask for a presentation
of W i.e. a list of all relations satisfied by the reflections in S.
Theorem 2.8. Given a set of simple reflections S, the group W has a presentation
W = 〈sα : α ∈ ∆ | (sαsβ)m(α,β) = 1 where the integers m(α, β) (3)
satisfy m(α, α) = 1, m(α, β) ≥ 2 for α 6= β. 〉. (4)
It’s natural to ask how our presentation would change if we changed the set of generators. An
arbitrary presentation might look quite different. However, if we ask that our generating set S
consists solely of reflections and |S| is as small as possible, then S corresponds to a set of simple
reflections. Moreover, Theorem 2.7 (3) implies that the presentation we get from two different
choices of simple roots is the same (up to a relabeling of the generators).
Remark 2.9. Groups that have a presentation as in (4) are called Coxeter groups since they were
extensively studied by him. We will see below the classification of finite Coxeter groups. However,
there is also a whole zoo of infinite Coxeter groups out there, which play a key role in several areas
of mathematics, such as geometric group theory.
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Example 2.10. If W is the dihedral group I2(m), then we can deduce from example 2.3 that
I2(m) = 〈sα, sβ | s2α = s2
β = (sαsβ)m = 1〉.
Similarly, for the symmetric group, example 2.4 implies that
Sn = 〈s1, . . . , sn−1 | s2i = 1, (sisj)
2 = 1, ∀|i− j| > 1, (sisi+1)3 = 1, i = 1, . . . , n− 2〉
The relations (sisj)2 = 1 and (sisi+1)3 = 1 can be rewritten as
sisj = sjsi, ∀|i− j| > 1, sisi+1si = si+1sisi+1, ∀ i = 1, . . . , n− 2.
They are called the braid relations since they are the defining relations of the braid group Bn, of
which Sn is a quotient.
2.2 Coxeter graphs and symmetric forms
So far we have used the notion of root system and simple roots to show that a finite reflection
group has a very particular presentation. Our approach to classifying these groups will be to try
and classify the finite groups that have these particular presentations. In order to do this, we need
some background on symmetric forms and symmetric matrices.
A symmetric bilinear form (−,−) on Rn is the same1 as a real n × n-matrix A such that
AT = A i.e. A is a symmetric matrix. Explicitly, given a form (−,−), we fix a basis ε1, . . . , εn of
Rn and set
ai,j := (εi, εj),
so that (v, w) = vTAw for all v, w ∈ Rn.
Definition 2.11. A symmetric real matrix A is said to be
• positive definite if xTAx > 0 for all 0 6= x ∈ Rn.
• positive semi-definite if xTAx ≥ 0 for all x ∈ Rn.
Before we relate reflection groups to symmetric matrices, we will first construct a Coxeter
graph out of W . Then there is a simple algorithm to associate to the Coxeter graph a symmetric
bilinear form (or rather, the algorithm shows that the Coxeter graph “remembers” the fact that
1Rather, it is the same as a real symmetric matrix once a basis of Rn has been chosen.
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it came from a reflection group acting on Rn, with its standard form). So we fix a set of simple
roots ∆ for R. The vertices of the Coxeter graph Γ are labeled by the corresponding simple roots
/ simple reflections. Associated to any pair α, β ∈ ∆, we have the integer m(α, β) ≥ 2, as in
Theorem 2.8. If m(α, β) = 2 then there is no edge between α and β. Otherwise, we add an edge
between α and β, decorated with the number m(α, β). Since m(α, β) = 3 occurs frequently, we
don’t bother to label the edge in this case.
Example 2.12. Going back to our examples of the dihedral and symmetric groups, example 2.10
implies that the Coxeter graph for I2(m) ism
. For the symmetric group Sn+1, we get
the Coxeter group of type An,
with n vertices.
In general a Coxeter graph is a finite graph Γ with at most one edge between two vertices, no
loops (i.e. an edge from a vertex to itself) and an integer m ≥ 3 labeling each edge. Next, if Γ has
n vertices, then we associated to it a symmetric bilinear form on Rn, or equivalently a symmetric
real matrix A. Let A = (aα,β), where aα,α = 1 and
aα,β = − cosπ
m(α, β), ∀ α 6= β.
Proposition 2.13. Assume that the real symmetric matrix A comes from a finite reflection group
as above. Then A is positive definite.
Proof. Let (−,−) denote the usual symmetric bilinear form on Rn such that the usual basis
{ε1, . . . , εn} is orthonormal. Then it is clearly positive definite. The matrix A doesn’t depend of
the length of the simple roots so we may assume without loss of generality that ||α|| = 1 for all
α ∈ ∆. As noted in section 2.5 below, (α, β) = ||α||||β|| cos θ, where θ is the angle between α and
β. Theorem 2.7 (4) implies that θ ≥ π. We want to express θ in terms of m(α, β). The key is the
fact that sαsβ is rotation by twice θ (see the exercise sheet). This implies that θ = π(
1m(α,β)
+ 1)
and hence
(α, β) = cos π
(1
m(α, β)+ 1
)= − cos
π
m(α, β).
This means that αTAβ = (α, β) for all α, β ∈ ∆ i.e. A is just the standard form (−,−) expressed
in the basis of Rn given by the simple roots ∆. In particular, it is positive definite.
What we will see from the classification is that the finite reflection groups, up to isomorphism,
are in bijection with the Coxeter graphs whose associated symmetric matrix A is positive definite.
14
For brevity, we will say that Γ is a positive (semi-)definite Coxeter graph if the corresponding
matrix A is positive (semi-)definite.
Exercise 2.14. Show that the symmetric matrix(1 − cos π
m
− cos πm
1
)
corresponding to the Coxeter graphm
is positive definite.
2.3 The classification
There are two steps in the classification of positive definite Coxeter graphs. First, we guess all
positive definite and semi-definite graphs. A direct calculation shows whether a particular graph
is positive (semi-)definite. Then we use this list, together with a key result from linear algebra to
prove that there are no others.
The proof of the following proposition is a direct calculation. By induction, one calculates the
eigenvalues of the associated symmetric matrix A (give it a go!).
Proposition 2.15. The following Coxeter graphs are positive definite.
An BCn4
Dn
E6 E7
E8 F44
G26
H35
H45
I2(m)m
Similarly, one checks by a direct calculation:
Proposition 2.16. The following Coxeter graphs are all positive semi-definite.
15
An
Bn4
Cn4 4
Dn
E6 E7
E8 F4
4G2
6
We recall the key “Spectral Theorem for real symmetric matrices”. A proof of the theorem
can be found in the appendix.
Theorem 2.17. Let A be a real valued symmetric matrix.
1. The eigenvalues of A are real.
2. There exists an orthogonal matrix g ∈ O(n,R) such that gAgT is diagonal.
Example 2.18. The spectral theorem for real symmetric matrices implies that a real symmetric
matrix is positive (semi-)definite if and only if all its eigenvalues are > 0 (≥ 0). As an example,
consider the Coxeter graph of type F4. The corresponding symmetric matrix is
A =
1 −1
20 0 0
−12
1 −12
0 0
0 −12
1 −1√2
0
0 0 −1√2
1 −12
0 0 0 −12
1
.
A rather lengthy computation shows that the eigenvalues of A are 0, 12, 1, 3
2and 2. Therefore F4
is positive semi-definite.
16
A matrix A is said to be decomposable if, after a permutation of rows and columns, it is the
sum of two blocks
A =
(A1 0
0 A2
),
otherwise we say that A is indecomposable. We will use the spectral theorem to prove the following
result in linear algebra. The proof of the following key proposition is based on Theorem 5.1. The
proof can be found on [12, page 35].
Proposition 2.19. Let A be a symmetric real matrix which is
(a) positive semi-definite,
(b) indecomposable,
(c) satisfies ai,j ≤ 0 for all i 6= j.
Then the radical
radA = {v ∈ Rn | vTAv = 0}
of A equals the kernel of A, and is one-dimensional. Moreover, the smallest eigenvalue of A
has multiplicity one, and the corresponding eigenvector can be chosen to have all strictly positive
coordinates.
Corollary 2.20. If Γ is a connected semi-positive definite Coxeter graph, then every proper sub-
graph is a positive definite Coxeter graph.
Proof. If Γ is connected then the matrix A is indecomposable and clearly satisfies the other
conditions of Proposition 2.19. Let Γ′ be a subgraph of Γ and A′ the corresponding symmetric
matrix. If |Γ′| = k ≤ |Γ| = n, then A′ is a k × k matrix. Then m′(α, β) ≤ m(α, β), which implies
that a′α,β = − cos πm′(α,β)
≥ aα,β = − cos πm(α,β)
= aα,β. We will assume that Γ′ is not positive
definite (in particular, Γ′ is a proper subgraph of then there exists some non-zero vector v ∈ Rk
such that vTA′v ≤ 0. If v′ = (|v1|, . . . , |vk|, 0, . . . , 0) then
0 ≤∑α,β
aα,β|vα||vβ| ≤∑α,β
a′α,β|vα||vβ|
≤∑α,β
a′α,βvαvβ ≤ 0.
Here the sum is over all α, β ≤ k and we have used the fact that aα,β ≤ 0 if α 6= β. Therefore
the inequalities are equalities. Moreover, we see that (v′)TAv′ = 0. Therefore Proposition 2.19
17
implies that v′ is in the kernel of A. But again, Proposition 2.19 implies that k = n and each vi is
non-zero. However, the other inequalities now imply that a′α,β = aα,β for all α, β i.e. A′ = A and
hence Γ′ = Γ, contradicting the fact Γ′ is a proper subgraph of Γ.
Theorem 2.21. Let Γ be a finite connected graph. The positive definite Coxeter forms listed in
Proposition 2.15 constitute all such forms.
Proof. The key to the proof of this theorem is Corollary 2.20. By Corollary 2.20, if we are given a
positive semi-definite graph then every proper graph is positive definite. We will show that every
positive semi-definite graph Γ is contained in the list given in Proposition 2.16. It will then follow
that the list given in Proposition 2.15 contains all positive definite Coxeter graphs.
We will need the following fact. An easy calculation of eigenvalues shows that
Z45
Z55
are not positive (semi-)definite graphs.
So we assume that Γ is a positive semi-definite Coxeter graph. Recall that the valence of a
vertex is the number of edges leaving the vertex.
(a) Let us consider first cycles in Γ. If Γ contains a cycle of length at least three, then it contains
An for some n. Hence it must equal An. So we assume Γ contains at most 2-cycles.
(b) If Γ contains a pair of vertices with ≥ 5 edges (or in our presentation, this is an edge labeled
by m ≥ 7) between them then the fact that it cannot contain G2 implies that Γ = I2(m)
for some m. But then it is not positive semi-define, so there are at most 4 edges between
vertices.
(c) If Γ contains an edge of weight 6, then there must be at least three vertices in Γ or else
Γ = I2(6). But if it contains at least three vertices it contains a copy of G2 (and must equal
G2). Hence we may assume that every edge of Γ has weight at most 5.
(d) Assume that Γ has an edge of weight 5. If it has only three vertices then the other edge
cannot be of weight 4 since Γ would contain a copy of B2. So this edge must have weight 3.
But then Γ = H3 is positive definite. Thus, Γ must have at least 4 vertices.
(e) The fact that Γ cannot contain Z4 implies that the unweighted graph is of type A such that
5 labels the left most edge.
18
(f) The fact that Γ cannot contain Z5 implies that Γ has exactly 4 vertices.
(h) The fact that Γ cannot contain C3 implies that Γ must in fact be H4. But then Γ is positive
definite. So we conclude that there are no positive semi-define Coxeter graphs with an edge
of weight 5.
(i) Thus, we are reduced to the situation where every edge of Γ has weight at most 4. If Γ
contains at least two edges of weight 4 then it contains Cn. So we may assume that Γ
contains at most one edge of weight 4.
(j) If the unweighted graph of Γ contains a vertex of valence at least 3 then it contains a copy
of Bn. Therefore we may assume that the unweighted graph of Γ is of type A.
(k) If Γ has at least 5 vertices then it contains a copy of F4 or is of type BCn, so it must equal
F4.
(l) If Γ contains fewer than 5 vertices then it is either F4 or type BCn i.e. it is not positive
semi-definite. Hence we are finally reduced to the case where every edge has weight 3.
(m) If Γ has a vertex of valency at least 4 then it contains a copy of D4. So we may assume that
each vertex has valency at most 3.
(n) If it contains at least two vertices of valency 3 then it contains a copy of Dn. Therefore we
may assume it contains exactly one vertex of valency 3.
(o) Then Γ = T (p, q, r) is the spoke with arms of length p, q, r (so that Γ has p+ q + r edges in
total).
(p) At least one of p, q, r is equal to 1 otherwise Γ contains a copy of E6 or is of positive definite
type. So without loss of generality, p = 1.
(p) If q, r ≥ 3 then Γ contains a copy of E7. Therefore we may assume without loss of generality
that q = 2.
(q) If r ≥ 4 then Γ contains a copy of E8, otherwise it is of positive definite type.
The fact that the positive definite graphs listed in Proposition 2.15 are all possible ones is proved
in a similar fashion, but the proof is much simpler. We leave it to the interested reader.
19
We haven’t quite completed the classification of finite reflection groups. One still has to
construct an explicit reflection group for each of the positive definite Coxeter graphs of Proposition
2.15. For the infinite series An,BCn and Dn this is straight-forward (we have seen type A already,
type BC corresponds to the hyperoctahedral group and D is the normal subgroup of BCn of index
2 consisting of all signed permutation matrices such that the product of the non-zero entries is
one), but the exceptional reflection groups of type E,F and H are rather difficult to construct. We
have actually seen the reflection group H3 already - this is a reflection group acting on R3, it is
none other than W (D), the symmetry group of the dodecahedron.
2.4 Crystallographic groups
Most reflection groups W have the additional property that they preserve a lattice L ⊂ Rn i.e.
sα(L) ⊂ L for all α ∈ ∆. Here a lattice is the Z-span Z · β1 + · · ·Z · βn of some basis {β1, . . . , βn}of Rn. In this case we say that W is a crystallographic reflection group.
Lemma 2.22. If W is crystallographic then m(α, β) = 2, 3, 4 or 6 for all α 6= β.
Proof. Assume that W preserves a lattice L = Z·β1+· · ·Z·βn. Then, any element w ∈ W , written
as a matrix with respect to the basis {β1, . . . , βn} has trace in Z. Now, sαsβ is rotation by 2πm(α,β)
(see exercise 3 (c)), hence Tr(sαsβ) = (n − 2) + 2 cos 2πm(α,β)
which implies that cos 2πm(α,β)
∈ 12Z.
This forces m(α, β) = 2, 3, 4 or 6.
Based on this simple observation, it is natural to extend the definition of root system.
Definition 2.23. Let R be a root system. Then R is said to be crystallographic if, in addition to
axioms (R1) and (R2) of definition 2.2, we have
(R3) If α, β ∈ R then
nβ,α :=2(α, β)
(α, α)
belongs to Z.
We can easily check from the classification of Theorem 2.21 which finite reflection groups have
the property that m(α, β) = 2, 3, 4 or 6 for all α 6= β. These are the ones of type A,BC,D,E,F
and G. Then, for each of these, one can explicitly construct a crystallographic root system. It
turns out that for all types, except BC, there is only one crystallographic root system of that type.
However, for the reflection group of type BC, there are two different crystallographic root systems,
which we call type B and type C. This construction implies:
Proposition 2.24. If W is crystallographic then it admits a crystallographic root system.
20
β
α
A1 × A1 β
α
A2
Figure 4: The crystallographic root systems of type A1 × A1 and A2.
2.5 The angle between roots
Recall that if α, β ∈ E are non-zero vectors, then the angle θ between α and β can be calculated
using the cosine formula ||α|| · ||β|| cos θ = (α, β). Thus,
〈β, α〉 =2(β, α)
(α, α)= 2||β||||α||
cos θ,
and hence 〈β, α〉〈α, β〉 = 4 cos2 θ. If α, β ∈ R, then the fourth axiom implies that 4 cos2 θ is a
positive integer. Since 0 ≤ cos2 θ ≤ 1, we have 0 ≤ 〈β, α〉〈α, β〉 ≤ 4. Hence the only possible
values of 〈α, β〉 are:
〈β, α〉 〈α, β〉 θ
0 0 π2
1 1 (?)
−1 −1 2π3
1 2 (??)
−1 −2 3π4
1 3 π6
−1 −3 (? ? ?)
(5)
From this, it is possible to describe the rank two root systems. They are of type A1 × A1,A2,B2
and G2. See Figures 2.5 and 2.5. In each of the diagrams, {α, β} form a set of simple roots.
2.6 The surfer dude and E8
The crystallographic root system of type E8, which is by far the most complicated of all the root
systems, made world headlines in 2007. First, there was the story of how a group of mathematicians
21
α
β
B2
α
β
G2
Figure 5: The crystallographic root systems of type B2 and G2. The long roots are coloured redand the short roots are black.
had finally computed the “Kazhdan-Lusztig-Vogan polynomials” for the reflection group of E8 (the
group W (E8) has order 696729600), using some clever computer algorithms and a huge number of
computers; the computation was even picked up in the main-stream media. See
http://aimath.org/E8/
http://www.aimath.org/E8/computerdetails.html
For details on the mathematics involved, check out
http://www-math.sp2mi.univ-poitiers.fr/~maavl/pdf/Nieuw-Archief.pdf
too.
Then, a short while later, a paper appeared on the internet by Garrett Lisi, claiming that
the physical “Grand Unifying Theory of Everything” i.e. the ultimate theory of physics, which
would finally combine general relativity with quantum mechanics, could be constructed using the
representation theory of the Lie group of type E8. This outlandish idea actually seemed to have
some physical and mathematical merit and was intensely studied by both groups for a short while.
What made the story even more interesting was that Lisi was actually a professional surfer, based
in Hawaii. Though he had a PhD in physics, he had left academia for surfing. As far as I am
aware his idea didn’t work out in the end, but you can dig into this yourself, starting with
http://www.telegraph.co.uk/news/science/large-hadron-collider/3314456/
22
Surfer-dude-stuns-physicists-with-theory-of-everything.html
https://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything
http://www.scientificamerican.com/article/garrett-lisi-e8-theory/
Finally, the beautiful Figure 2.6 is the projection of the E8 root system (which lives in R8) onto
the plane.
2.7 Invariant Theory
Finite reflection groups appear in many situations in mathematics (and physics). One of the most
classical appearances is in invariant theory. Let W ⊂ GL(Rn) be a group. Then the group W also
acts naturally on the polynomial ring R[x1, . . . , xn] by (g·f)(a) = f(g−1·a) for all f ∈ R[x1, . . . , xn],
g ∈ W and a ∈ Rn. The ring R[x1, . . . , xn]W of functions invariant under W is a very complicated
ring in general. The study of these rings is called invariant theory. One of the cornerstones of the
theory is the Chevalley Theorem.
Theorem 2.25. Assume W ⊂ GL(Rn) is finite. Then R[x1, . . . , xn]W is a polynomial ring if and
only if W is a reflection group.
Example 2.26. We can make the symmetric group Sn act on Rn by permuting the coordinates.
We’ve seen that this makes Sn into a reflection group. Then R[x1, . . . , xn]Sn is the ring of sym-
metric polynomials. It is a classical result (attributed to Newton) that
R[x1, . . . , xn]Sn = R[e1, . . . , en],
where
er(x1, . . . , xn) =∑
1≤i1<···<ir≤n
xi1 · · ·xir
is the rth elementary symmetric polynomial.
Exercise 2.27. Take Z2 = {1, s} acting on R2 by s · x1 = −x1 and s · x2 = −x2. Show that
R[x1, x2]Z2 is not a polynomial ring.
2.8 Lie algebras
Finite crystallographic reflection groups, described via root systems, first appeared in a (seemingly)
very different context.
23
Figure 6: The root system of E8.
24
At the opposite end of the spectrum to reflection groups are groups of continuous transforma-
tions; for instance, the group of all possible rotations of the plane. These groups have an additional
structure, namely the set of elements in the group is a smooth space or manifold and the multi-
plication map G×G→ G is a differentiable map (as is the inverse map). These groups are called
Lie groups, after the ground breaking work of the Norwegian mathematician Sophus Lie. These
groups are often very complicated as spaces. Therefore, in order to understand them better, the
standard approach is to replace them by a linearization - a vector space approximation of G, with
some additional structure that “remembers” some of the structure of the group. This additional
structure on the vector space g is a bilinear map [−,−] : g× g→ g that makes (g, [−,−]) into a
Lie algebra. Formally, the axioms of a Lie algebra are
(L1) The bracket [−,−] is anti-symmetric i.e.
[X, Y ] = −[Y,X], ∀ X, Y ∈ g.
(L2) The bracket [−,−] satisfies the Jacobi identity i.e.
[X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z,X]] = 0, ∀ X, Y, Z ∈ g.
What does all this have to do with root systems? Lie algebras are “built up”, in a precise sense,
from the simple Lie algebras. Remarkably, one can associate, in a very natural way, to every
simple Lie algebra a crystallographic root system. It was shown by E. Cartan and Killing that a
Lie algebra is uniquely defined by its root system. Thus, the classification of simple Lie algebras
is equivalent to the classification of root systems.
The typical example of a Lie algebra is the space g = Matn(R) of all n×n real matricies. The
Lie bracket on g is the commutator bracket [A,B] := AB−BA. This is the Lie algebra associated
with the Lie group GL(Rn) of all invertible linear maps Rn → Rn; or in more down to earth terms,
the group of all invertible n× n real matricies.
Exercise 2.28. Check that (Matn(R), [−,−]) satisfies the axioms of a Lie algebra.
Lie groups and Lie algebras come under the general heading of Lie theory. It is a beautiful
and incredibly rich theory, which I would strongly recommend to everyone.
2.9 Remarks
There is a huge body of literature on the very classical subject of finite reflection groups and
general Coxeter groups. Two standard references are the books Reflection Groups and Coxeter
25
Groups, by Humphreys [12], and Geometry of Coxeter groups, by Hiller [10]. A quick google will
reveal thousands more references.
26
3 Quivers
In the finally two lectures, we turn instead to quivers and their representations. Though these
don’t seem to have anything to do with reflection groups, we’ll see in the final lecture on Gabriel’s
Theorem that reflection groups and their root systems play a key role in the representation theory
of quivers.
3.1 What is a quiver?
A quiver Q is simply a directed graph. That is, it is a collection Q0 = {e1, . . . , ek} of vertices and
a collection Q1 = {α1, . . . , α`} of arrows between vertices. In order to specify which two vertices
an arrow α goes between, we define a pair of functions h, t : Q1 → Q0 such that h(α) is the vertex
at the head of α and t(α) is the tail of α i.e. α is the arrow beginning at vertex t(α) and ending
at vertex h(α).
Example 3.1. Here are some examples of quivers.
e2
e1 e2 e1 e5 e3 e1
e4
α
α β
γ
α
δ
e1 e1 e2
α1
α2
···
αn
α
β
For instance, in the second example, Q0 = {e1, e2, e3, e4, e5}, Q1 = {α, β, γ, δ} and t(γ) = e3,
h(β) = e5 etc.
Remark 3.2. A quiver is the case, holdall, or tube that an archer uses to hold his/her arrows.
3.2 Representations
As representation theorists, we’re interested in representations of a quiver. A representation
M = {(Vi, ϕα) |i ∈ Q0, α ∈ Q1} is the data of a complex vector space Vi at each vertex i of Q
27
and a linear map ϕα : Vt(α) → Vh(α) between the vector space at the tail of α to the vector space
at the head of α, for every arrow α.
Example 3.3. Here are two representations
M : C2 C3, N : C4 Cϕα
ϕβ
φα
φβ
(6)
where
ϕα =
4 2
3 3
0 9
, ϕβ =
−4 34
0 0
12 π
, φα =(
1 2 3 4), φβ =
(4 1 −101 19
).
We can associate to a representation M = {(Vi, ϕα)} a dimension vector dim M ∈ Z|Q0| by
setting
(dim M)i = dimVi.
Unlike a vector space, the representation M is not uniquely defined by its dimension vector.
However, as we shall see, dim M is a very useful invariant of M .
Example 3.4. If we consider the two representations M and N in (6), then
dim M = (2, 3), dim N = (4, 1).
3.3 Subrepresentations, simple representations etc.
Given a representation M = {(Vi, ϕα) | i ∈ Q0, α ∈ Q1} of Q, a subrepresentation N of M is the
representation {(Ui, ϕα|Ut(α)) | i ∈ Q0, α ∈ Q1}, where each Ui is a subspace of Vi, chosen so that
ϕα(Ut(α)) ⊂ Uh(α) for all α ∈ Q1.
Example 3.5. Here is an example of a subrepresentation. LetM be the representation of e1 e2
α
β
given in (6). Let U1 = C be the subspace of C2 spanned by v1 =
(1
−1
)and U2 = C2 the sub-
space of C3 spanned by
u1 =
1
0
0
, u2 =
0
0
1
.
28
Then, M restricts to the subrepresentation U1 U2
ϕ′α
ϕ′β
, where
ϕ′α =
(2
−9
), ϕ′β =
(−38
12− π
).
If on the other hand, we take the representation N in (6) and U1 the subspace of C4 spanned by
the vectors
u1 =
1
0
0
0
, u2 =
0
1
0
0
,
and U2 = {0} ⊂ C, then this does not produce a subrepresentation of N since
φα(U1) 6⊂ U2, φβ(U1) 6⊂ U2,
i.e. one cannot just choose arbitrary subspaces to get subrepresentations.
Example 3.6. If we consider the example of Q being e1α
then a representation of Q is
simply a vector space V1 together with a linear map ϕα : V1 → V1. Then a subspace W1 ⊂ V1 is a
subrepresentation precisely if ϕα(W1) ⊆ W1.
We say that the representation M is simple if the only subrepresentations of M are M itself
and 0. Here is a construction of some simple representations. For each i ∈ Q0, we let E(i) =
{(E(i)j, ϕα)} denote the representation where E(i)i = C and E(i)j = 0 for j 6= i. Moreover,
ϕα = 0 for all α. We leave it to the reader to check that E(i) is simple.
Given two representations M = {(Vi, ϕα)} and N = {(Ui, ψα)} we can construct a new repre-
sentation M ⊕ N , the direct sum of M and N , as follows. At each vertex i, we simply take the
direct sum of vector spaces Vi ⊕ Ui. Then, we define the new map ηα = (ϕα, ψα) for each α ∈ Q1
i.e.
ηα =
(ϕα 0
0 ψα
): Vt(α) ⊕ Ut(α) −→ Vh(α) ⊕ Uh(α).
29
Notice that M and N are naturally subrepresentions of the direct sum M ⊕N .
Of course, to better understand representations of a given quiver, we want to be able to compare
them. The standard way of doing this is to consider homomorphisms between representations.
This basic notion is very important in the theory. See the exercises for more details.
3.4 The path algebra
There is a natural algebra we can associate to each quiver Q. Recall that an algebra A is a C-
vector space together with a bilinear multiplication map A×A→ A that makes it into a ring with
identity. First we define a path in Q to be a tuple of arrows αkαk−1 · · ·α1 such that h(αi) = t(αi+1)
for i = 1, . . . , k − 1. The length of the path is k. We think of each vertex ei as being a path of
length zero i.e. the path that does nothing. Let CQ be the complex vector space with basis given
by all paths in Q. To define a multiplication on CQ, it suffices to do so for paths. Then,
(βl · · · β1) ◦ (αk · · ·α1) =
{βl · · · β1αk · · ·α1 if h(αk) = t(β1)
0 otherwise.
That is, multiplication is simply the concatenation of paths where-ever that makes sense. For the
paths ei of length zero, we have
(αk · · ·α1) ◦ ei =
{αk · · ·α1 if t(α1) = i
0 otherwise.
Example 3.7. The path algebra CQ of
Q : e1 e2
α
β
has basis {e1, e2, α, β} and multiplication is given by
e2 · α = α · e1 = α, e2 · β = β · e1 = β, α · β = β · α = 0, . . .
But notice that the path algebra CQ′, where Q′ : e1α
has basis {e1, α, α2, α3, . . . } and
30
multiplication
e1 · αn = αn · e1 = αn, αn · αm = αn+m.
In particular, CQ′ is infinite dimensional and commutative.
In group theory, it is often very advantageous to study the way that groups can act on other
objects such as sets, vector spaces, manifolds... Similarly, to better understand an algebra, it is very
advantageous to study actions of the algebra on vector spaces. This is the field of representation
theory. At its heart is the notion of a module over the algebra A.
Definition 3.8. An A-module M is a vector space equipped with a bilinear map A ×M → M ,
(a,m) 7→ a ·m such that
• 1 ·m = m for all m ∈M .
• a · (b ·m) = (a · b) ·m for all a, b ∈ A and m ∈M .
We will be interested, in particular, in the category of modules for the path algebra CQ.
Exercise 3.9. Notice that the definition of a module uses explicitly the identity element 1 of the
algebra A. However, we’ve made no mention of an identity for the path algebra. Check that
1 :=∑i∈Q0
ei
is the identity in CQ.
Example 3.10. IfQ is the quiver e1 e2
α
β
then we could takeM = Cf1⊕Cf2 with ei·fj = δi,jfj
and
α · f1 = 4f2, α · f2 = 0, β · f1 = 12f2, , β · f2 = 0,
or N = C7, where e1 · v = v and e2 · v = α · v = β · v = 0 for all v ∈ C7.
3.5 Representations vs. CQ-modules
We have seen that we can associated to a quiver its category of representations and also the
category of modules over the path algebra. It turns out that these are equivalent notions (formally,
one says that the categories are equivalent). To show that they are equivalent, we explain how to
go from a representation to a module for the path algebra and back.
31
Let M = {(Vi, ϕα) | ei ∈ Q0, α ∈ Q1} be a representation of Q. We construct out of M a
module F (M) for the path algebra CQ. As a vector space,
F (M) :=⊕ei∈Q0
Vi,
is just the direct sum of the spaces Vi. Since every path in CQ is the composition of a series of
arrows, we just have to say how a) the trivial paths ei act on F (M) b) how each of the arrows
α ∈ Q1 acts on F (M). Firstly, for each i we have a projection map pi : F (M) → Vi and an
inclusion map qi : Vi ↪→ F (M). We define
ei = qi ◦ pi i.e. ei · v = qi ◦ pi(v).
Next, associated to α ∈ Q1 is the linear map ϕα : Vt(α) → Vh(α). We use a similar trick to extend
this to a map F (M)→ F (M) by setting
α = qh(α) ◦ ϕα ◦ pt(α) : F (M)qh(α)−→ Vt(α)
ϕα−→ Vh(α)
pt(α)−→ F (M),
i.e. α · v = qh(α) ◦ ϕα ◦ pt(α)(v).
Now we want to go in the opposite direction. Given a module N for the path algebra CQ,
we want to define a representation G(N) = {(Ui, ψα) | ei ∈ Q0, α ∈ Q1}. Notice first that the
identity element 1 in CQ is the sum∑
i∈Q0ei. Also, the rules for mutiplication in CQ show that
ei · ej = δi,jei (we say that {ei ∈ Q0} form a complete set of orthogonal idempotents in CQ). This
implies that
N =⊕i∈Q0
ei ·N.
We set Ui = ei · N . We just need to define a linear map ψα : et(α)N → eh(α)N for each α ∈ Q1.
For this, notice that
Theorem 3.11. The maps F and G define inverse equivalences between the category of all rep-
resentations of Q and the category of all CQ-modules.
Proof. We have done all the hard work already in defining the maps F and G. It is straight-
forward to check that G ◦ F (M) = M for any representation M and F ◦ G(N) = N for any
CQ-module N .
Example 3.12. Let’s take Q to be the quiver e1 e2 e3α
βand the representation N :
C C C3
6. Then CQ has basis {e1, e2, e3, α, β} and F (N) is the three-dimensional
32
module with basis f1, f2, f3 such that
ei · fj = δi,j, α · f1 = 3f2, β · f3 = 6f2,
and α · fi = β · fj = 0 otherwise. Similarly, if we take the CQ-module M in example 3.10 then
G(M) is the representation of Q given by
C C.4
7
Remark 3.13. In modern terms, the above theorem should really be phrased in the language of
category theory. The problem with this technical language is that it hides completely the very
simple idea behind Theorem 3.11. None the less, I’ll give the extra details for those of you of a
more abstract nature. Firstly, we are really considering the categories Rep(Q) of representations
of Q and CQ-Mod of left CQ-modules. This is not only the data of the set (or rather class)
Ob(Rep(Q)) of all representations, but also for each pair of representations M,N ∈ Ob(Rep(Q)),
the vector space HomQ(M,N) of all morphisms M → N . Similarly, the objects Ob(CQ-Mod) of
CQ-Mod are the modules and HomCQ(M,N) is the vector space of morphisms of CQ-modules.
Then F and G are functors between these two categories i.e. they are defined not only on the
objects but also on the hom-spaces of these categories,
FM,N : HomQ(M,N)→ HomCQ(F (M), F (N)), f 7→ F (f),
in a way that is compatible with compositions of morphisms. Finally, Theorem 3.11 says that F
and G are quasi-inverse equivalences of categories.
3.6 Remarks
There is a huge body of literature on quiver representations, simply type “Quiver representations”
into google and see what you get. Here are some references [5], [15], [18], [8], [7], that I found
useful when learning about quivers.
33
4 Gabriel’s Theorem
In the previous lecture we were introduced to quivers and their representations. In particular, we
saw that the “smallest” possible representations are the simple ones and, given any two represen-
tations, we can make a new representation by taking their direct sum. Flipping this on its head,
we can ask if a representation can be decomposed into a direct sum of two smaller representations.
We say that a representation is indecomposable if it cannot be written as the direct sum of two
subrepresentations. In this final lecture, we’ll consider the question
Q. Which quivers have only finitely many indecomposable representations?
The proof of Gabriel’s Theorem is long and difficult, so to properly understand it you’ll need
to read several different presentations. In the remarks at the end of the section I’ve listed some
sources that contain a complete proof of the theorem.
4.1 Indecomposable representations
In order to study the representations of a quiver, the first thing to do is try and break a repre-
sentation up into a direct sum of simpler representations - we say that M decomposes as a direct
sum M1 ⊕M2. We’ve already seen that many representations occur as the direct sum of smaller
representations. Repeating, we get smaller and smaller representations. Of course this process
must eventually stop. That is, eventually we get to a representation M that cannot be written as
the direct sum M1 ⊕M2 of two subrepresentations.
Definition 4.1. The representation M is said to be indecomposable if it cannot be decomposed
into a direct sum of two proper subrepresentations.
Notice that we can always decompose M = M ⊕ 0, but such a decomposition doesn’t help us
understand M better, so we ignore these trivial decompositions. Clearly, if we want to understand,
or classify all representations of Q, it suffices to classify the indecomposable representations. In
general this turns out to be a very difficult (in fact, impossible; see section 4.5) task. It is because
of this that it’s so important to know which quivers only have finitely many indecomposables.
4.2 Gabriel’s Theorem
Here is the statement of Gabriel’s Theorem. The proof is rather long, but it will bring together
ideas from representation theory, geometry, reflection groups and combinatorics in a very clever
way.
34
Theorem 4.2 (Gabriel). 1. The quiver Q admits finitely many indecomposables (up to iso-
morphism) if and only if the underlying graph is of type A, D or E.
2. In the case where Q has underlying graph a positive definite Euler graph, M 7→ dim M
defines a bijection between the isomorphism classes of indecomposable representations and
the positive roots R+ of the corresponding root system.
Part (2) means that each indecomposable M of Q is uniquely defined, up to isomorphism, by
its dimension vector and given any v ∈ R+ one can construct an indecomposable representation
M of Q such that dim M = v. This latter statement is the really difficult part of Gabriel’s
theorem.
Remark 4.3. There is a much more general result, called Kac’s Theorem; see [13]. This theorem
gives a precise description of the set of dimension vectors of the indecomposable representations
for any quiver Q. Moreover, Kac defined “real” and “imaginary” dimension vectors, and showed
that there is only one indecomposable representation of Q with dimension vector v if v is a real
vector and infinitely many indecomposable representations of Q with dimension vector v if v is
an imaginary vector. In this more general setting, Gabriel’s Theorem is saying that every root is
real if the underlying graph of Q is positive definite.
4.3 The proof part I: orbits
In this section we prove the easier direction of Gabriel’s Theorem: if Q has only finitely many
indecomposable representations then the underlying graph must be of type A, D or E. Firstly, we
introduce the space Rep(Q,v) of all representations with dimension vector v. If we have vector
spaces of dimension m and n respectively, then the space Hom(Cm,Cn) of all linear maps from
Cm to Cn has dimension m × n. Now, to define a representation M of Q with dimension vector
v = (v1, . . . , vk) we choose, for each α ∈ Q1, an element ϕα ∈ Hom(Cvt(α) ,Cvh(α)). Thus, M is a
point in
Rep(Q,v) =⊕α∈Q1
Hom(Cvt(α) ,Cvh(α)).
The space Rep(Q,v) is a vector space of dimension∑α∈Q1
vt(α)vh(α).
What does it mean for two representations M,N ∈ Rep(Q,v) to be isomorphic? Well, if M =
{(Cvi , ϕα)} and N = {(Cvi , ψα)} it means that there are invertible linear maps fi ∈ GL(Cvi) for
35
each i ∈ Q0 such that the diagrams
Cvt(α) Cvh(α)
Cvt(α) Cvh(α)
ϕα
ft(α) fh(α)
ψα
commute for all α ∈ Q1. That is, ψα = fh(α) ◦ ϕα ◦ f−1t(α). Define an action of the group G(v) :=∏
i∈Q0GL(Cvi) on the vector space Rep(Q,v) by
f · (ϕα) = (fh(α) ◦ ϕα ◦ f−1t(α)), ∀ f ∈ G(v).
Then we have shown that:
Lemma 4.4. The two representations M,N ∈ Rep(Q,v) are isomorphic if and only if N is in
the G(v)-orbit of M .
Now we’ll need some basic facts from algebraic geometry (this is the study of spaces defined
by systems of polynomial equations). These spaces are called algebraic varieties e.g. the elliptic
curve C = {(x, y) | y2 = x3 + x2} ⊂ C2 is an algebraic variety. Just as for a vector space, these
spaces also have “dimension”; the dimension of C is one, as you might expect. Now the key fact
we’ll use is that the G(v)-orbits in Rep(Q,v) are not just sets, they are also algebraic varieties!
Therefore, we can compute the dimension of an orbits G(v) ·M := {f ·(ϕα) | f ∈ G(v)}. Moreover,
there’s a geometric analogue of the classical orbit-stabilizer theorem; see [5, Proposition 2.1.6] for
a proof.
Proposition 4.5. Let M ∈ Rep(Q,v). Then,
dimG(v) ·M = dimG(v)− dim StabG(v)(M).
What is the dimension of G(v)? Well, the set (in fact, variety) of all invertible linear maps
GL(Cn) is an open subset of all linear maps Hom(Cn,Cn). This implies that dimGL(Cn) =
dim Hom(Cn,Cn) = n2. Therefore,
dimG(v) = dim∏i∈Q0
GL(Cvi) =∑i∈Q0
dimGL(Cvi) =∑i∈Q0
v2i .
36
Define the Ringel form 〈−,−〉 on ZQ0 by
〈v,w〉 =∑i∈Q0
viwi −∑α∈Q1
vt(α)wh(α). (7)
Eventually, we want to arrive at a positive definite symmetric form, just as in the classification of
finite reflection groups. However, the Ringel form 〈−,−〉 is not symmetric. Therefore, the Euler
form is defined to be the symmetrization of the Ringel form,
(v,w)E := 〈v,w〉+ 〈w,v〉
= 2∑i∈Q0
viwi −∑α∈Q1
vt(α)wh(α) + wt(α)vh(α).
It does not depend on the orientation of Q i.e. it only depends on the underlying graph. But it
is important to note that this form we have just associated to the underling graph of Q is not the
Coxeter form! To distinguish between the two different symmetric forms, we will refer to a graph
as an Euler graph when we mean that the associated form is the Euler form. Similarly, a graph Γ
will be said to be a positive (semi-)definite if the associated Euler form is positive (semi-)definite.
Why is the Euler form useful? Let’s compute:
1
2(v,v)E =
∑i∈Q0
v2i −
∑α∈Q1
vt(α)vh(α), (8)
= dimG(v)− dim Rep(Q,v) (9)
= dimG(v) ·M + dim StabG(v)(M)− dim Rep(Q,v), (10)
where we have used Proposition 4.5 in the last step.
Lemma 4.6. If Q has only finitely many indecomposable representations then (v,v)E ≥ 2 for all
v ∈ NQ0.
Proof. If there are only finitely many indecomposable representations of Q, then for a fixed di-
mension vector v ∈ NQ0 there are only finitely many representation of Q up to isomorphism with
dimension vector v. Geometrically, this means that there are only finitely many G(v)-orbits in
Rep(Q,v). Since the dimension of the union of finitely many algebraic varieties is the maxi-
mum of the dimension of the varieties, there must be an open orbit in Rep(Q,v) i.e. an orbit
G(v) ·M such that dimG(v) ·M = dim Rep(Q,v). For this orbit, equation (10) implies that12(v,v)E = dim StabG(v)(M). Thus, we just need to show that dim StabG(v)(M) ≥ 1. For λ ∈ C×,
37
consider the element λ = (λ IdCvi ) ∈ G(v). Then
λ · (ϕα) = (λ−1ϕαλ−1) = (ϕα)
i.e. there is a diagonal copy of C× in StabG(v)(M) for any representation M . Since dimC× = 1,
this implies that dim StabG(v)(M) ≥ 1, as required.
Just as in the classification of finite reflection groups, we are aiming to reduce the classification
problem of Gabriel’s Theorem to one of classifying symmetric bilinear forms constructed from
finite graphs.
Proposition 4.7. Let Q be a quiver with only finitely many indecomposable representations. Then
the Euler form (−,−)E is positive definite.
Proof. We need to show that (v,v)E > 0 for all non-zero w ∈ Rn. If there exists a non-zero
vector w ∈ Rn such that (v,v)E ≤ 0 then there exists w ∈ QQ0 such that (v,v)E ≤ 0. Rescaling,
we may assume that w ∈ ZQ0 . Finally, notice from the explicit formula (7) for (v,v)E that if
|w| := (|w1|, |w2|, . . . , ) ∈ NQ0 then
(|w|, |w|)E ≤ (v,v)E ≤ 0.
Thus, we have shown that if (−,−)E is not positive definite then there exists a non-zero vector
w ∈ NQ0 such that (v,v)E ≤ 0. We deduce from Lemma 4.6 that (−,−)E is positive definite.
The proof of the following proposition is a direct, inductive, calculation.
Proposition 4.8. The following Euler graphs are positive definite.
An Dn
E6 E7 E8
The following Euler graphs are positive semi-definite.
An Dn
38
E6 E7 E8
Now we can state the classification theorem for Euler graphs that implies the easy direction
in Gabriel’s Theorem.
Theorem 4.9. If Γ is a finite connected graph whose Euler form is positive (semi-)definite, the
Γ appears on the list given in Proposition 4.8.
Proof. As you can see from exercise 4.11, it is possible to deduce the theorem from Theorem 2.21.
Here we will give a direct proof.
Just as in the proof of Theorem 2.21, the key result is Corollary 2.20, though the analysis
is much simpler in this case. We will concentrate on classifying the positive definite graphs (we
do not actually need Corollary 2.20 for this), the argument for positive semi-definite graphs is
similar. So we assume that Γ is positive definite. This means that no subgraph of Γ can be
positive semi-definite. Recall that the valence of a vertex is the number of edges leaving the
vertex. Hence,
(a) Γ contains no cycles (An),
(b) Each vertex of Γ has valence at most 3 (D4),
(c) There is at most one vertex with valence 3 (Dn),
(d) If every vertex has valence at most 2 then Γ is type A,
(e) We assume that there is one vertex of valence 3. Then Γ = T (p, q, r) is a spoke with arms
of length p, q, r (so that Γ has p+ q + r edges in total).
(f) At least one of p, q, r is equal to 1 (E6). So without loss of generality, p = 1.
(g) q ≤ 2 (E7). If q = 1 then Γ is of type D. Thus, we assume q = 2.
(h) r ≤ 4 (E8) and hence Γ is of type E.
39
Exercise 4.10. Using Corollary 2.20 and the arguments of the proof of Theorem 4.9, show that
the positive semi-definite Euler forms listed in Proposition 4.8 constitute all such forms.
Exercise 4.11. You’ll notice that the positive definite Euler graphs are precisely the positive definite
Coxeter graphs that are simply laced i.e. have at most one edge between any two vertices. Let
(−,−)C , resp. (−,−)E, be the Coxeter form, resp. the Euler form, associated to a graph Γ.
1. Show that if Γ is simply laced then (−,−)E = 2(−,−)C .
2. If Γ is not simply laced, show that there is no λ ∈ R such that (−,−)E = λ(−,−)C .
3. Show that the symmetric matrix (2 −m−m 2
)
corresponding to the Euler graphm
is positive definite if and only if m = 1. When
is it positive semi-definite?
4. By considering the subgraphsm
with m > 1 of Γ, show that a non-simply laced
Euler graph is not positive definite.
5. Deduce Theorem 4.9 from Theorem 2.21.
4.4 The proof part II: reflection functors
In order to relate the representation theory of Q to the corresponding root system, we will recall
the proof, in terms of reflection functors, given by Bernstein, Gelfand and Ponemarev. The
presentation we give here is based on unpublished lecture notes by H. Derksen.
Since we have defined the Euler form (−,−)E on ZQ0 , we can define, for each loop free vertex
i ∈ Q0, a reflection si by
si(v) = v − (v, ei)ei.
Then, if w = si(v), we have wj = vj for j 6= i and
wi = −vi +∑h(α)=i
vt(α) +∑t(α)=i
vh(α).
In this way we get a reflection group W (Q) acting on ZQ0 .
40
Remark 4.12. In general, W (Q) will be an infinite group. It will be the same as the Coxeter group
associated to the underlying graph of Q, as defined in Theorem 2.8, if and only if Q is simply
laced. One can prove this using the results of exercise 4.11.
We call a vertex i ∈ Q0 a sink if there are no arrows emanating from i i.e. there is no α ∈ Q1
such that t(α) = i. Similarly, i is said to be a source if there are no arrows ending in i i.e. there is
no α ∈ Q1 such that h(α) = i. For each sink or source i of Q, we will define a reflection functor at
i that, given a representation M of Q, will produce a new representation whose dimension vector
is si(dim M). This will allow us to construct all indecomposable representations of Q starting
from the simple representations E(i).
First, we assume that i is a sink. We define a functor S+i : Rep(Q)→ Rep(siQ), where siQ is
the quiver obtained by reversing all the arrows heading into i (so that i becomes a source in siQ).
Let M = {(Vj, ϕα)} be a representation of Q. Then S+i (M) = {(Uj, ψα)}, where Uj = Vj for all
j 6= i, and if α ∈ Q1 with h(α), t(α) 6= i, then ψsi(α) = ϕα. At i, we define Ui to be the kernel of
the map ⊕h(α)=i
Vt(α) → Vi, (vt(α)) 7→∑h(α)=i
ϕα(vt(α)).
If h(α) = i then t(si(α)) = i in siQ and we define
ψsi(α) : Ui → Uh(si(α)) = Vt(α)
to be the restriction to Ui of the projection map⊕
h(α)=i Vt(α). The functor S+i acts on morphisms
between representations in the obvious way.
Next we consider the case where i is a source. We define a functor S−i : Rep(Q)→ Rep(siQ).
Again, let M = {(Vj, ϕα)} be a representation of Q. Then S−i (M) = {(Uj, ψα)}, where Uj = Vj
for all j 6= i, and if α ∈ Q1 with h(α), t(α) 6= i, then ψsi(α) = ϕα. At i, we define Ui to be the
cokernel of the map
Vi →⊕t(α)=i
Vh(α), vi 7→ (ϕα(vi)).
If t(α) = i then h(si(α)) = i in siQ and we define
ψsi(α) : Ut(si(α)) = Vh(α) → Ui
to be the composition of the embedding Vh(α) ↪→⊕
t(β)=i Vh(β) followed by the quotient map⊕t(β)=i Vh(β) → Ui. Just as for sinks, it is straight-forward to define the functor S−i on morphisms
between representations.
41
Remark 4.13. What we have done is take an algebraic structure, in this case a reflection si, and
“lift” it to a functor S+i between categories. Thus, the categories Rep(Q) and Rep(siQ) are
playing the role of ZQ0 . This notion of lifting algebraic structures, in particular representations,
to categories and functors between them is called categorification. It is a relative new concept, but
one that has turned out to be incredibly powerful. As such, a great deal of research is currently
devoted to “categorifying” representations. For more on this fascinating subject, see [14] and [16].
The following theorem is the key result needed to complete the proof of Gabriel’s Theorem.
The proof is long and involved, so it won’t be given here. For the details see the references given
at the end of the section.
Theorem 4.14 (Bernstien-Gelfand-Ponomarev). Let i ∈ Q0 be a sink and M an indecomposable
representation of Q. Either,
1. M = E(i) and S+i (E(i)) = 0; or
2. S+i (M) 6= 0 is indecomposable S−i S
+i (M) 'M and dim S+
i (M) = si(dim M).
Let i ∈ Q0 be a source and M an indecomposable representation of Q. Either,
1. M = E(i) and S−i (E(i)) = 0; or
2. S−i (M) 6= 0 is indecomposable S+i S−i (M) 'M and dim S−i (M) = si(dim M).
From now on, we assume that the underlying graph of Q contains no cycles (this is no big
restriction since we ultimately only want to consider the quivers of type A,D or E). Then we
can relabel the vertices Q0 = {1, . . . , n} of Q so that t(α) < h(α) for all α ∈ Q1. We define the
Coxeter element in W (Q) as
c = s1 · · · sk.
Its inverse is c−1 = sk · · · s1. A moment’s thought shows that cQ = Q since each arrow is reversed
exactly twice. This means that we can lift c and c−1 to functors
C+ := S+k ◦ · · · ◦ S
+1 : Rep(Q)→ Rep(Q)
and
C− := S−1 ◦ · · · ◦ S−k : Rep(Q)→ Rep(Q).
Theorem 4.14 now implies
Corollary 4.15. Let M be an indecomposable representation of Q.
42
1. Either C+(M) = 0 or C+(M) is an indecomposable representation such that dim C+(M) =
c(dim M).
2. Either C−(M) = 0 or C−(M) is an indecomposable representation such that dim C−(M) =
c−1(dim M).
From now until the end of the proof of Gabriel’s Theorem, we assume that Q is of type A,D
or E. Then, as noted in exercise 4.11, the two symmetric forms (−,−)E and (−, )C agree and we
can think of ZQ0 as a subset of Rn by sending the dimension vectors ei to the basis of Rn given
by the simple roots ∆ = {e1, . . . , en}. Since the underlying graph of Q is of crystallographic type
(see section 2.4), we may assume that the root system ∆ ⊂ R is of crystallographic type. Notice
that we have a preferred set of positive roots R+ ⊂ NQ0 since we have specified ∆. The reflection
group W (Q) acts on R and Theorem 2.7 implies that W (Q) ·∆ = R i.e each α in R can be written
(non-uniquely) as w(ei) for some w ∈ W (Q) and ei ∈ ∆.
Lemma 4.16. Let W be a reflection group of type A,D or E and α ∈ NQ0 a non-zero dimension
vector. Then c(α) 6= α and there exists some k � 0 such that ck(α) /∈ NQ0.
Proof. For the first part of the statement, we require the following identity
〈α, c(β)〉 = −〈β, α〉, ∀ α, β ∈ ZQ0 . (11)
Unfortunately, the proof of this equation uses the notion of projective and injective representations,
but if you’d like to see the proof see Lemma 4.4.1 and Lemma 4.4.2 of [15]. Assume that there
exists some α 6= 0 such that c(α) = α. Then equation (11) implies that
(α, α) = 〈α, c(α)〉+ 〈α, α〉 = −〈α, α〉+ 〈α, α〉 = 0.
But we have shown that the Euler form is positive definite for Q. Therefore this implies that
α = 0; a contradiction.
For the second part, notice that c has finite order since W is finite. Therefore, if the order of c
is n say, the root α+ c(α) + · · ·+ cn−1(α) is invariant under c, hence it is zero. This implies that
some ck(α) does not belong to NQ0 .
Finally we can proof the hard part of Gabriel’s Theorem.
Theorem 4.17. Let Q be a quiver, whose underlying graph is of type A,D or E. Then there is
an indecomposable of dimension α if and only if α ∈ R+. Moreover, any such indecomposable is
unique, up to isomorphism.
43
Proof. Let M be an indecomposable representation of Q with dimension vector α. By Lemma
4.16, there exists some k > 0 such that ck(α) /∈ NQ0 . This means that (C+)k(M) = 0. Choose k
minimal such that (C+)k(M) = 0. Then, if N = (C+)k−1(M), we have N 6= 0 and C+(N) = 0.
Then Corollary 4.15 implies that there is some i such that S+i ◦ · · · ◦ S+
1 (N) = E(i+ 1) is simple
with dimension vector ei+1. It also implies that
M = (C−)k−1(N) = (C−)k−1 ◦ S−1 ◦ · · · ◦ S−i (E(i+ 1)).
This shows that α = ck−1s1 · · · si(ei+1) is a positive root and M is uniquely defined by its dimension
vector.
Finally, we just need to show that if β is an arbitrary positive root, then there exists an
indecomposable M with dimension vector β. Again, by Lemma 4.16, we can choose k > 0 such
that ck(β) /∈ NQ0 but ck−1(β) ∈ NQ0 . Corollary 4.15 implies that sisi−1 · · · s1(β) = ei+1 for some
i. In this case, we can take M = (C−)k−1 ◦ S−1 ◦ · · · ◦ S−i (E(i + 1)). Corollary 4.15 ensures that
M is an indecomposable with dimension vector β.
Remark 4.18. For an alternative proof using the geometry of orbit closures, see [7] or [5].
4.5 Finite, tame and wild
What about the positive semi-definite Euler graphs? Do their quiver representations have some
particularly interesting properties? The answer is yes, but it order to give you the correct state-
ment, I need to tell you something about tame and wild quivers. We know already that if Q is
not of positive definite type then there will be infinitely many indecomposable representations
up to isomorphism. Even if we fix a dimension vector v, this will still be the case. However, in
many examples one sees that we can still parameterize the indecomposables as an infinite family
of modules. For instance, if we consider the quiver 1 2α
β
with dimension vector (1, 1) then
the indecomposables are in bijection with points on the projective line P1. We say that Q is
tame if, for any fixed dimension vector, there are at most finitely many one-parameter families of
indecomposable representations of Q. Otherwise Q is said to be wild. Kac’s Theorem says that Q
is tame if and only if the underlying graph is a positive semi-definite Euler graph. In the above
example, the underlying graph is of type A2. For details on this, see [1, Section 4.4].
Example 4.19. The following example is very classical and can be viewed as one of the motivating
example behind the definition of a quiver. Fix a vector space Cn and consider the problem of
classifying k-tuples of subspaces (V1, . . . , Vk) with dimVi = ni ≤ n. We don’t care about change of
44
basis in Cn, so we think of (V1, . . . , Vk) as equivalent to (U1, . . . , Uk) if there is some g ∈ GL(Cn)
such that
(U1, . . . , Uk) = (gV1g−1, . . . , gVkg
−1).
We can encode this problem as a problem about quiver representations. Let Q be the spoke quiver
with one central vertex 0 and k surrounding vertices 1, . . . , k with an arrow αi from each outer
vertex i to 0. When k = 4, we have
1
2 0 4
3
α1
α2
α4
α3
If we take the dimension vector v = (n, n1, . . . , nk), then classifying the subspaces is the same as
classifying representations of Q with dimension vector v such that each ϕαi is injective - a subset
of all representations. Then Gabriel’s Theorem says that there are only finitely many collections of
subspaces when k ≤ 3. When k = 4, we get the Euler graph D4, which is positive semi-definite by
Proposition 4.8. Therefore, for each v there are finitely many 1-dimensional families of collections
of subspaces. When k > 4, the classification problem is wild.
4.6 The universality of Coxeter/Euler graphs
Behind much of what we have seen already is a Coxeter/Euler graph of some type or other. Recall
that the set of positive definite Coxeter graphs are the ones of type A,B,D,E,F,G,H and I. The
graphs of type A,B,D,E,F and G are the crystallographic Coxeter graphs and the simply laced
ones (those of type A,D and E) are precisely the positive definite Euler graphs.
Each of the following classes of objects are classified by Coxeter, or Euler, graphs of some
sort. In each case, this is because the classification problem is shown to be equivalent to the
classification of Coxeter/Euler graphs of various types.
• The classification of finite reflection groups. We’ve seen that these are classified by positive
definite Coxeter graphs.
• The classification of quivers of finite type. Gabriel’s Theorem shows that these are classified
by positive definite Euler graphs.
45
• The classification of simple Lie algebras. This (essentially2) reduces to the problem of
classifying crystallographic Coxeter graphs; see [11, Chapter 11].
• The classification of rational surface singularities. These are classified by positive definite
Euler graphs; see [17].
• The classification, up to conjugation, of finite subgroups of SU(2). As explained in [20], the
McKay correspondence shows that these groups are also classified by positive definite Euler
graphs.
• The positive definite Euler graphs also classify the “elementary catastrophes” appearing in
Thom’s Catastrophe theory. See section 9 of [19] for details.
In this list there is no mention of the original objects we started with, the Platonic solids. If
you are willing to be a bit flexible with your definitions then they also fit into this classification.
The “dihedron” is defined to be the regular p-gon but with an infinitesimal thickening to make
it a 3-dimensional solid. It has Schlafli symbol {p, 2} for p ≥ 2. Similarly, the “orange” is the
regular solid where each face only has two edge (think of Terry’s chocolate orange). It has Schlafli
symbol {2, q} for q ≥ 3. In this way we get
• The classification of Platonic solids is given by positive definite Euler graphs.
• There is a bijection between the Platonic solids and the finite subgroups of SO(3,R) (i.e.
the finite rotationl groups), up to isomorphism. This is given by sending a solid P to its
group W0(P ) of rotational symmetries. Thus, the finite rotation groups are classified by
positive definite Euler graphs.
4.7 Where to next?
What we’ve covered so far is just the tip of a huge, wonderfully complex, iceberg. We’ve seen
how finite reflection groups, root systems, Lie algebras and quiver representations are all related
by Coxeter and Euler graphs. These are just some of the many mathematical objects that have
these graphs underlying them. Some others are listed in section 4.6 above. Each of these objects
is a particular example of a rich story of it’s own. For instance rational surface singularities are
the some of the most basic examples of singular algebraic varieties in algebraic geometry. This
2Strictly speaking they are classified by the corresponding Dynkin diagram, which also encodes the relativelength of the roots of the crystallographic root system.
46
is a vast subject, with a long history, but still the focus of attention for many mathematicians
worldwide. A similar statement can be made about Lie theory.
I would just like to mention two key areas that today play a role on the interface between
representation theory and geometry and that I think are particularly beautiful generalizations of
the above theory. The first of these is the generalized McKay correspondence - the original McKay
correspondence relates the representation theory of finite subgroups G of SU(2) with the geometry
of the resolution of the corresponding quotient singularity C2/G. Generalizations of this aim to
relate resolutions of the singularities Cn/G, for G ⊂ SU(n), with the representation theory of G.
The most general result in this direction is the generalized McKay correspondence of Bridgeland,
King and Reid [4]. A closely related result was shown by Bezrukavnikov and Kaledin [3].
The second is the theory of Nakajima’s quiver varieties. These are moduli spaces of represen-
tations of quivers, satisfying some additional relations. They can be used for instance to construct
resolutions of the quotient singularities C2/G. But they also appear in the representation theory
of certain infinite dimensional Lie algebras called Kac-Moody Lie algebras, are ubiquitous in the
construction of categorifications, and have applications in geometry and theoretical physics. For
an introduction to quiver varieties, have a look at the lecture notes [9] by Ginzburg.
4.8 Remarks
For a crash course on Gabriel’s Theorem and Kac’s Theorem, try the very well written article
[8]. The extensive notes [15] also contain a detailed proof of Gabriel’s Theorem via reflection
functors (as well as many other interesting applications of these functors). The original paper [2]
by Bernstein-Gelfand-Ponomarev, where they prove Gabriel’s Theorem using reflection functors
is also very readable - a copy is available at
http://www.math.tau.ac.il/~bernstei/Publication_list/
publication_texts/BGG-CoxeterF-Usp.pdf
Other notes that contain a proof of Gabriel’s Theorem include [5], [18], and [7].
47
5 Appendix
5.1 The Spectral Theorem for symmetric matrices
The Spectral Theorem for real valued symmetric matrices is a very useful result. We include a
proof here.
Theorem 5.1. Let A be a real valued symmetric matrix.
1. The eigenvalues of A are real.
2. There exists an orthogonal matrix g ∈ O(n,R) such that gAgT is diagonal.
Here O(n,R) = {g ∈ GL(n,R) | g−1 = gT} is the real orthogonal Lie group. You might think
that it’s not so surprising that a matrix with real coefficients has real eigenvalues. However, most
matrices with real coefficients actually have complex eigenvalues. As a simple examples, consider(0 1
−1 0
),
(2 5
−3 6
),
(0 −7
1 12
).
The second part of the the spectral theorem is saying that one can always find an orthonormal
basis {u1, . . . , un} of Rn i.e. a basis such that (ui, uj) = δi,j, such that each ui is an eigenvector
for A.
Proof of Theorem 5.1. First we note that A is symmetric if and only if (Au, v) = (u,Av) for all
u, v ∈ Rn; it is enough to check this when u = ei and v = ej are standard basis elements, then
(Au, v) = aj,i and (u,Av) = ai,j.
As noted above, (2) is equivalent to showing that there is an orthonormal basis {u1, . . . , un}such that each ui is an eigenvector of A (with eigenvector αi say). Therefore, we will construct the
basis {u1, . . . , un} by induction on n and show that each αi is real. The case n = 1 is vacuous since
A is real valued. Thus, we may assume by induction that the theorem holds for all symmetric
(n− 1)× (n− 1) matrices.
In general, a real valued matrix need not have any real eigenvectors. So the first (and in
fact hardest) step is to show that A has a real eigenvector with real eigenvalue. To do this, we
briefly need to step into the complex world where we are guaranteed the existence of at least one
eigenvector. A complex valued matrix B is called unitary if B† = B, where B† :=(B)T
= (BT )
and B means take the complex conjugate of every entry of B. Clearly every real symmetric matrix
48
is unitary when considered as a complex matrix. Let u be an eigenvector for A (over C) with
eigenvalue λ. We wish to show that λ is real. If u† := uT then u† · u =∑n
i=1 |ui|2 ∈ R>0. Then
(u†Au) = λ(u† · u) = λ(u†u).
On the other hand,
(u†Au) = u†A†u
= u†Au = λ(u† · u),
where the first equality is true for any matrix and the second is because A is unitary. Hence we
deduce that (λ− λ)(u† · u) = 0. This means that λ = λ is real. If we write u = v +√−1w, where
v, w ∈ Rn, then
Au = (Av) +√−1(Aw) = (λv) +
√−1(λw).
Therefore, both v and w are real eigenvectors for A with eigenvalue λ. Since u 6= 0, at least one
of them is non-zero. Let un be this real eigenvector. Rescaling if necessary, we may assume that
(un, un) = 1. Set αn := λ.
Now, we’ll use the induction hypothesis to find the vectors u1, . . . , un−1. Let
V = u⊥n := {v ∈ Rn | (un, v) = 0},
an (n−1)-dimensional subspace of Rn. Since A is symmetric and Aun = αnun, we have (Av, un) =
αn(v, un) = 0 for all v ∈ V , which means that A(V ) ⊂ V . Chose an arbitrary basis for V and
write A′ for the (n− 1)× (n− 1) matrix of A restricted to V . Then
A =
(A′ 0
0 αn
).
Moreover, (A′u, v) = (Au, v) = (u,Av) = (u,A′v) for all u, v ∈ V which means that A′ is also
a symmetric real matrix. By induction there exists an orthonormal basis {u1, . . . , un−1} of V
such that A′ui = αiui. Then {u1, . . . , un−1, un} is the basis of Rn that we have been looking for
(check!).
49
5.2 Rotations in R3
In this section we will prove that
The transformation g ∈ GL(R3) is a rotation if and only if g ∈ SO(3,R). (12)
Here SO(3,R) is the group of all matrices g ∈ GL(R3) such that det(g) = 1 and ggT = 1. A
rotation of R3 is a transformation that fixes pointwise a line and acts as a rotation in the plane
perpendicular to that line.
We will prove the above in a series of easy claims. We begin with something more general.
A complex valued matrix B ∈ GL(Cn) belongs to the unitary group U(n) if B−1 = B†, where
B† :=(B)T
= (BT ) and B means take the complex conjugate of every entry of B. Clearly
O(n,R) ⊂ U(n).
Lemma 5.2. Let A ∈ U(n).
1. There exists g ∈ U(n) such that gAg† is diagonal.
2. The eigenvalues λi of A have modulus |λi| equal to 1.
Proof. We prove both claims by induction on n, just as in the proof of Theorem 5.1. The case
n = 1 is vacuous.
We extend the usual inner product (−,−) on Rn to the Hermitian form 〈−,−〉 on Cn given
explicitly by
〈z, w〉 =n∑i=1
ziwi.
Then A ∈ U(n) if and only if 〈Az,Aw〉 = 〈z, w〉 for all z, w ∈ Cn.
Since we are working with complex matrices there is at least one eigenvector z ∈ Cn, with
eigenvalue λ say, for A i.e. Az = λz. Rescaling, we may assume that ||z||2 = 〈z, z〉 = 1. Let
V = z⊥ = {x ∈ Cn | 〈x, z〉 = 0}. Then the fact that A ∈ U(n) implies that A(V ) ⊂ V . Set un = z
and λn = λ. As in the proof of Theorem 5.1, if A′ is the restriction of A to V , then A′ ∈ U(n− 1)
and
A =
(A′ 0
0 λn
).
By induction, A′ can be diagonalized and all eigenvalues of A′ have modulus 1. Moreover, the
corresponding eigenvectors u1, . . . , un−1 are orthonormal with respect to 〈−,−〉 Let us check that
|λn| = 1. We have
1 = 〈z, z〉 = 〈Az,Az〉 = 〈λz, λz〉 = λλ〈z, z〉 = |λ|2.
50
Finally, we should check that there is some g ∈ U(n) such that gAg† is the diagonal matrix with
diagonal entries λ1, . . . , λn. We just take g to be the matrix with columns u1, . . . , un. If e1, . . . , en
is the standard orthonormal basis of Cn, then
〈gei, gej〉 = 〈ui, uj〉 = δi,j = 〈ei, ej〉
which implies that g ∈ U(n). By construction, gAg† is the required diagonal matrix.
In the case A ∈ O(n,R) ⊂ U(n), the fact that the coefficients of the characteristic equation
det(A− tId) are real implies that if λ is an eigenvalue for A, so too is its complex conjugate. Thus,
if A ∈ O(3,R), there exists g ∈ U(3) and θ ∈ R such that
gAg† =
±1 0 0
0 e√−1θ 0
0 0 e−√−1θ
.
In particular, if A ∈ SO(3,R) then A has an eigenvalue equal to one. If it has more than one
eigenvalue equal to one, then A will be the identity matrix. Therefore, without loss of generality,
the real matrix Id − A has a one-dimensional kernel ` ⊂ R3. Let u1 ∈ R3 span the kernel.
Rescaling if necessary, we may assume that ||u1|| = 1. Since A is orthogonal, (Av,Aw) = (v, w)
for all v, w ∈ R3. In particular, this implies that A(`⊥) ⊂ `⊥, where `⊥ = {v ∈ R3 | (u1, v) = 0}.Fix an orthonormal basis u2, u3 of `⊥. Then {u1, u2, u3} is an orthonormal basis of R3. With
respect to this basis,
A =
(1 0
0 A′
),
where A′ ∈ SO(2,R). The only thing left to show is that A′ acts as a rotation of the plane R2
spanned by {u2, u3}. But it is an easy direct calculation to show that B ∈ GL(R2) is a rotation
if and only if B ∈ SO(2,R). This completes the proof of (12).
Finally, we note:
Proposition 5.3. Let P be a Platonic solid. Then W (P ) is a finite subgroup of O(3,R).
Proof. We have already seen that W (P ) is a finite group. Rescaling P if necessary, we may
assume that the vertices of P all lie in the 3-sphere S3 = {v ∈ R3 | ||v|| = 1}. Notice that every
transformation g ∈ W (P ) preserves S3. Therefore it suffices to show that the symmetry group
W (S3) of S3 equals O(3,R). It’s just as easy to consider W (Sn) ⊂ GL(Rn). Since O(n,R) can
51
be defined as those g ∈ GL(Rn) such that (g · v, g · w) = (v, w) for all v, w ∈ Rn, we see that
||g ·v|| = ||v|| = 1 for all v ∈ S3. Thus, O(n,R) ⊂ W (Sn). To show the opposite inclusion, it suffices
to show that if u, v are a pair of orthonormal vectors in Rn i.e. (v, v) = (u, u) = 1, (u, v) = 0, then
(g · u, g · u) = (g · v, g · v) = 1 and (g · u, g · v) = 0 for all g ∈ W (Sn). The first two conditions are
automatic. To show that (g · u, g · v) = 0, we expand
2 = (u− v, u− v) = (g · (u− v), g · (u− v))
= (g · u, g · u) + (g · v, g · v)− 2(g · u, g · v)
= 2− 2(g · u, g · v)
Thus, (g · u, g · v) = 0 as required.
52
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